The role of solid solutions in iron phosphate-based electrodes for selective electrochemical lithium extraction

Electrochemical intercalation can enable lithium extraction from dilute water sources. However, during extraction, co-intercalation of lithium and sodium ions occurs, and the response of host materials to this process is not fully understood. This aspect limits the rational materials designs for improving lithium extraction. Here, to address this knowledge gap, we report one-dimensional (1D) olivine iron phosphate (FePO4) as a model host to investigate the co-intercalation behavior and demonstrate the control of lithium selectivity through intercalation kinetic manipulations. Via computational and experimental investigations, we show that lithium and sodium tend to phase separate in the host. Exploiting this mechanism, we increase the sodium-ion intercalation energy barrier by using partially filled 1D lithium channels via non-equilibrium solid-solution lithium seeding or remnant lithium in the solid-solution phases. The lithium selectivity enhancement after seeding shows a strong correlation with the fractions of solid-solution phases with high lithium content (i.e., LixFePO4 with 0.5 ≤ x < 1). Finally, we also demonstrate that the solid-solution formation pathway depends on the host material’s particle morphology, size and defect content.

. Different intercalation C rates (0.01C, 0.1C, 0.2C, or 0.5C; 0.1C equals to 14.7 mA/g) were used until 70% of the total capacity. N2 (purity > 99.998%) was continuously bubbled into the solution to avoid side reactions caused from dissolved O2. (Noting that χ 2 gives an estimation of the distance between the real data and the simulated data. Its expression is:

Supplementary
with ( ) is the measured impedance at the frequency; 7 mA/g) intercalation curve in 500 mL 1 mM LiCl: 1 M NaCl aqueous solution and C/30 (4.9 mA/g) de-intercalation curve in 60 mL 30 mM NH4HCO3 recovery solution, with the use of 70% of the total capacity (102.9 mAh/g). For both intercalation and de-intercalation, N2 (purity > 99.998%) was continuously bubbled into the solution to avoid side reactions caused from dissolved O2, and the reference electrode is Ag|AgCl|KCl (4.0 M) with the testing temperature at room temperature (20 ~ 25 °C). NaFePO4 was used as the counter electrode during the intercalation process; while graphite rod was used as the counter electrode during the de-intercalation process.    (Noting that CPE1 = 1 1( ) 1 , CPE2 = 1 2( ) 2 , = −1/2 − −1/2 and the error is calculated using the Levenberg-Marquardt algorithm, which can be assimilated to a standard deviation. It gives an estimate of the relevancy of the parameter. If the error is very high it means that a great variation of the parameter will not affect very much the quality of the fit. Hence, the considered parameter is not critical in the minimization process.)

Total Energy Calculation
Total energies of structures were determined using DFT calculations with the project augmented-wave (PAW) 6 [11][12][13] . All structures were fully optimized until the energy was converged to within 10 -5 eV per supercell and the forces on each atom were less than 0.02 eV/Angstroms.

Structure Search
DFT energies of the LixNayFePO4 (0 ≤ x+y ≤ 1) system were fit using a cluster expansion (CE) model to search for low-energy configurations given a maximum supercell size. The CE formalism is a wellestablished approach for studying ordering in alloys [14][15][16][17] . In the CE model the mixing enthalpies of the structures are parametrized using clusters, . The mixing enthalpy of each structure's configuration is fit using a sum of weighted cluster correlation functions based on the products of occupation variables .
α is the effective cluster interaction (ECI) for the cluster . Using a chosen set of clusters, the energy of a structure with a configuration given by occupation variables is predicted using Eq. 1.
where is the multiplicity of cluster , which is determined by the symmetry of the parent lattice. In this study two cluster expansions were fit, one for the ternary system and a second one focusing on the Livacancy edge of the LixNayFePO4 system. In total, 506 DFT energies were calculated, with 161 of those on the Li-vacancy edge. The 506 structures show that the only intermediate structure stable with respect to the terminal compositions LiFePO4, NaFePO4 and FePO4 is Na0.66FePO4. From the set of 161 structures on the Li-vacancy edge, low energy configurations with greater separation of structural Li atoms and vacancies were selected for seven intermediate Li concentrations. The selected structures were later used to calculate the difference between Li and Na intercalation potentials. The search for low energy configurations considered all supercells containing at most 86 atoms. In this work, the ICET package was used for the construction of the CE model 18 . A large cluster space (2280) with clusters up to the fourth order (quadruplets) were considered, and the Automatic relevance determination regression (ARDR) algorithm with regularization parameter, = 15000, was used to optimize a sparse set of clusters for the CE model.

Supplementary Note 2: Deconvolution of solid-solution fraction from diffraction patterns
To quantify the solid-solution fraction in Li-seeded FePO4, we fit the obtained X-ray diffraction patterns to a number of Gaussians, following previous work 19,20 Table 3. S4: Calculating (211) and (020) area ratios: The LeBail refinement of the FePO4 pattern showed that the ratio between the (211) and the (020) reflection areas is 0.38 (Supplementary Figure 4). Since the (211) peak and (020) peak in LiFePO4 are too close to distinguish, we use 0.23 as the area ratio according to the reference 20 . Area ratios of all intermediate phases also follow the linear combination of these two end-up phases, which are summarized in Supplementary Table 4. S5: Normalization of areas: The scattering factors of LiFePO4 and FePO4 differ. Therefore, all areas were normalized to the area of LiFePO4 by dividing the area of FePO4 by a factor of 1.24 20 . S6: Fitting the XRD spectra with nine species: two end phases, FePO4 and LiFePO4, and seven intermediate phases. The difference of state of charge (SOC) between two adjacent phases is set as 12.5%. And each phase will contribute two peaks, one is (020) peak and another one is (211) peak. So totally, we need to fit the band with 18 Gaussians. We can then get all the areas of peak (211)  peak.

Supplementary Note 3: Potential reasons for the deviations of calculated weighted sum of Li from fitting
As shown in Supplementary Figure 5 and Supplementary Table 5, we saw deviations in the calculated weighted sum of Li compared to our seeding range. We plotted the calculated weighted sum of Li from XRD fittings versus the electrochemically intercalated Li amount, as shown in Supplementary Figure 6. The relationship between the calculated weighted sum of Li and the depth of intercalation has good linearity (R 2 = 0.999). The deviation of the XRD fitted Li amount from electrochemical seeding amount indicates the possibility of unidentified system error. Even with error, this good linearity supports our analysis of the correlation of Li selectivity to Li solid solution phases since the error cannot be randomly affecting either the low-Li or high-Li solid solution phase fractions. Otherwise, there will not be a good linearity.
We think there could be two reasons that cause the deviations: 1 25 . In other words, the more accurate deconvolution of the XRD intensity band requires an infinite number of phases, which is assumed to be impossible and impracticable, and could also lead to overfitting issues. Since only nine different phases of state of charge were chosen to deconvolute the XRD patterns, this simplification assigned the other intermediate phases to the nine chosen ones, which could cause deviations of the calculated weighted sum of Li. 2. We find out that the intensity contributions from the polyvinylidene difluoride (PVDF), super P, and carbon cloth substrate could also introduce deviations by raising the background intensity, especially under the low depth of intercalation. PVDF and super P are indispensable binders and conductive additives for the preparation of FePO4 electrodes. The flexible carbon cloth is a good choice considering manufacturing and practical use. We then tried mirror polished glassy carbon with flatter surface as the substrate. The new glassy carbon substrate decreases the deviations caused by porous structures of carbon cloth. As shown in Supplementary Figure 7 and Supplementary Table 6, we achieved a better agreement between the calculated weighted sum of Li and the depth of intercalation, which also shows a good linear relationship in Supplementary Figure 8a. We also witnessed a similar monotonically increasing trend of high-Li SS phases with increased seeding range under 4C (588 mA/g) in Supplementary Figure 8b, while the low-Li SS phases still did not correlate with the increasing Li seeding range.
At this stage, although we witnessed some deviations of the calculated weighted sum of Li from deconvolution results, the general trend is repeatable and reasonable. Moreover, for the consistency of comparisons and practical feasibility, all the samples are tested on carbon cloth substrate unless specified.

Supplementary Note 4: Calculating Li-Na potential difference with respect to solvated ions
For the structures on the Li-vacancy edge of the LixNayFePO4 system the difference in potential for intercalating Li vs Na was calculated by determining the energy contribution from the electrochemical metal-ion insertion reaction shown in the equation below: where ∆ , ℎ is the energy to extract an isolated atom A from the cathode, IE is the ionization energy of A, and ∆ + is the solvation energy of A + . Precisely, according to the literature, the ionization energy for Li and Na are 5.39171 eV and 5.13908 eV respectively 27 . Meanwhile, the solvation energy for Li + and Na + are 5.389 eV and 4.198 eV respectively 28 . The Li and Na potentials were calculated at 8 different Liion concentrations (0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, and 0.875). At each composition, among the low energy configurations the configurations with greater separation of structural Li atoms and vacancies were selected. This is because the thermodynamically stable state of these compositions is a decomposition into LiFePO4 and FePO4. For each selected configuration a single Li or Na atom was placed on a vacancy site. If there were vacancy sites with different number of first, second, third structural Li nearest neighbors, two different calculations were performed. One where the added Li or Na atom was placed in order to maximize the proximity of structural Li atoms, and another where the Li or Na atom was placed as far away from the structural Li atoms as possible. Of the two resulting energies, the one with the lower energy was used for the Li and Na potential calculation. All configuration corresponded to 56 atom supercells, if all vacancies were filled, therefore with the addition of a single Li or Na atom x in the above equations is 0.125.
Then the Li-Na potential difference for each phase can be calculated using the following equation: The calculated Li-Na intercalation potential differences for each phase are summarized in Supplementary  Table 9. We also provided channel filling information for each intermediate phase in Supplementary Table  8 for reference. More negative Li-Na potential difference shows that the Li-ion intercalation is preferred to Na-ion intercalation.